Failure-recovery model with competition between failures in complex networks: a dynamical approach

Real systems are usually composed by units or nodes whose activity can be interrupted and restored intermittently due to complex interactions not only with the environment, but also with the same system. Majdand\v{z}i\'c $et\;al.$ [Nature Physics 10, 34 (2014)] proposed a model to study systems in which active nodes fail and recover spontaneously in a complex network and found that in the steady state the density of active nodes can exhibit an abrupt transition and hysteresis depending on the values of the parameters. Here we investigate a model of recovery-failure from a dynamical point of view. Using an effective degree approach we find that the systems can exhibit a temporal sharp decrease in the fraction of active nodes. Moreover we show that, depending on the values of the parameters, the fraction of active nodes has an oscillatory regime which we explain as a competition between different failure processes. We also find that in the non-oscillatory regime, the critical fraction of active nodes presents a discontinuous drop which can be related to a "targeted" k-core percolation process. Finally, using mean field equations we analyze the space of parameters at which hysteresis and oscillatory regimes can be found

Synchronization in interacting Scale Free Networks

We study the fluctuations of the interface, in the steady state, of the Surface Relaxation Model (SRM) in two scale free interacting networks where a fraction $q$ of nodes in both networks interact one to one through external connections. We find that as $q$ increases the fluctuations on both networks decrease and thus the synchronization reaches an improvement of nearly $40\%$ when $q=1$. The decrease of the fluctuations on both networks is due mainly to the diffusion through external connections which allows to reducing the load in nodes by sending their excess mostly to low-degree nodes, which we report have the lowest heights. This effect enhances the matching of the heights of low-and high-degree nodes as $q$ increases reducing the fluctuations. This effect is almost independent of the degree distribution of the networks which means that the interconnection governs the behavior of the process over its topology.Comment: 13 pages, 7 figures. Added a relevant reference.Typos fixe

Diffusion on a lattice: transition rates, interactions and memory effects

We analyze diffusion of particles on a two dimensional square lattice. Each lattice site contains an arbitrary number of particles. Interactions affect particles only in the same site, and are macroscopically represented by the excess chemical potential. In a recent work, a general expression for transition rates between neighboring cells as functions of the excess chemical potential was derived. With transition rates, the mean field tracer diffusivity, $D^\text{MF}$, is immediately obtained. The tracer diffusivity, $D = D^\text{MF} f$, contains the correlation factor $f$, representing memory effects. An analysis of the joint probability of having given numbers of particles at different sites when a force is applied to a tagged particle allows an approximate expression for $f$ to be derived. The expression is applied to soft core interaction (different values for the maximum number of particles in a site are considered) and extended hard core

Recovery of Interdependent Networks

Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. An important question that has not been addressed is how to repair a failing system before it suffers total breakdown. Here we introduce a recovery strategy of nodes and develop an analytic and numerical framework for studying the concurrent failure and recovery of a system of interdependent networks based on an efficient and practically reasonable strategy. Our strategy consists of repairing a fraction of failed nodes, with probability of recovery $\gamma$, that are neighbors of the largest connected component of each constituent network. We find that, for a given initial failure of a fraction $1-p$ of nodes, there is a critical probability of recovery above which the cascade is halted and the system fully restores to its initial state and below which the system abruptly collapses. As a consequence we find in the plane $\gamma-p$ of the phase diagram three distinct phases. A phase in which the system never collapses without being restored, another phase in which the recovery strategy avoids the breakdown, and a phase in which even the repairing process cannot avoid the system collapse

Reversible bootstrap percolation: Fake news and fact checking

Bootstrap percolation has been used to describe opinion formation in society and other social and natural phenomena. The formal equation of the bootstrap percolation may have more than one solution, corresponding to several stable fixed points of the corresponding iteration process. We construct a reversible bootstrap percolation process, which converges to these extra solutions displaying a hysteresis typical of discontinuous phase transitions. This process provides a reasonable model for fake news spreading and the effectiveness of fact checking. We show that sometimes it is not sufficient to discard all the sources of fake news in order to reverse the belief of a population that formed under the influence of these sources

Synchronization in Scale Free networks: The role of finite size effects

Synchronization problems in complex networks are very often studied by researchers due to its many applications to various fields such as neurobiology, e-commerce and completion of tasks. In particular, Scale Free networks with degree distribution $P(k)\sim k^{-\lambda}$, are widely used in research since they are ubiquitous in nature and other real systems. In this paper we focus on the surface relaxation growth model in Scale Free networks with $2.5< \lambda <3$, and study the scaling behavior of the fluctuations, in the steady state, with the system size $N$. We find a novel behavior of the fluctuations characterized by a crossover between two regimes at a value of $N=N^*$ that depends on $\lambda$: a logarithmic regime, found in previous research, and a constant regime. We propose a function that describes this crossover, which is in very good agreement with the simulations. We also find that, for a system size above $N^{*}$, the fluctuations decrease with $\lambda$, which means that the synchronization of the system improves as $\lambda$ increases. We explain this crossover analyzing the role of the network's heterogeneity produced by the system size $N$ and the exponent of the degree distribution.Comment: 9 pages and 5 figures. Accepted in Europhysics Letter

Design and Experimental Characterization of EDFA Based WDM Ring Networks with Free ASE Light Re-circulation and Link Control for Network Survivability

In this paper, we theoretically and experimentally investigate the performance of erbium-doped fiber amplifier (EDFA)-based WDM ring networks with free amplified spontaneous emission (ASE) light recirculation. We show that, with proper network and amplifier design, the lasing light generated by free ASE recirculation within the looped network provides an effective gain clamping technique, ensuring limited signal power excursions under WDM channels add-drop operations. Considering a ring network composed of eight fiber sections and eight EDFAs, maximum signal power overshoots below 2.5 dB have been measured under 23 24 WDM channels drop. Optical signal-to-noise ratio (OSNR) analysis and bit-error rate (BER) measurement at 10 Gb/s confirm acceptable performances and negligible penalties due to polarization effects and relative intensity noise transfer from laser light to WDM signals. We also propose and demonstrate a new link control technique which overcomes the main limiting factors of such networks, respectively, related to OSNR degradation, stability and survivability to fiber and EDFA breakages

Equilibrium Viscosity and Disequilibrium Rheology of a high Magnesium Basalt from Piton De La Fournaise volcano, La Reunion, Indian Ocean, France

Lava flows are a common hazard at basaltic to intermediate volcanoes and have posed a significant threat to La Reunion Island over the past centuries. In sustained flow units, the efficiency of lava transport away from the vent is dominated by cooling. For basaltic to intermediate lavas, it is the ability of the lava to solidify during cooling which exerts a first-order control on spatial extent and flow distance. As a consequence, understanding the sub-liquidus rheology of lavas has become a key focus in lava flow research in the past decade. To date, the development of a systematic understanding of lava rheology during emplacement conditions has been significantly hampered by a lack of experimental data. Here we present new data on the rheological evolution of crystallizing high-Mg basalt from Piton de la Fournaise. Sub-liquidus experiments were performed at constant cooling rates ranging from 0.5 to 5 K/min. Those rates mimic thermal conditions experienced 1) by lava during flow on the surface and 2) by magma during dike and sill emplacement. Our data show that the effective viscosity of the crystallizing suspension increases until reaching a specific sub-liquidus temperature, the so-called "rheological cutoff temperature" (T-cutoff), at which the lava becomes rheologically immobile and flow ceases. This departure from the pure liquid viscosity curve to higher viscosity is a consequence of rapid crystallization and its variability for a given lava is found to be primarily controlled by the imposed cooling rate. Based on these experimental data, we adapt the failure forecasting method (FFM) - commonly used to describe the self- accelerating nature of seismic signals to forecast material failure - to predict the rheological cut-off temperature (T-cutoff). The presented data substantially expand the modest experimental database on non-equilibrium rheology of lavas and represent a step towards understanding the underlying process dynamics